Math 5610 Fundamentals of Computational Mathematics: Homework 8 Problems

  1. Task: Implement a code that will approximate the largest eigenvalue of a matrix using the Power Iteration algorithm. Test the code in the Hilbert matrix. Include an entry in your software manual for the algorithm.
  2. Task: Implement the Inverse Iteration method with shifting to find approximate eigenvalues of a matrix A. Include an entry in your software manual documenting the algorithm. Find an eigenvalue less than the largest for the Hilbert matrix.
  3. Task: Write a code that will compute an approximation of the condition in the 2-norm using your code to produce a largest and smallest eigenvalue of a matrix. Document your algorithm in your software manual. Try the code on a Hilbert matrix.
  4. Task: Test the code developed above to compute the condition number of the Hilbert matrix as the size of the matrix is increased. Tabulate and/or graph the results from your work.
  5. Task: Write a code that will compute approximations of the smallest and largest eigenvalues. Next subdivide the interval containing the smallest and largest eigenvalues and use the points in the subdivision to shift and locate other eigenvalues. Discuss your results.
  6. Task: Implement the Raleigh Quotient algorithm for computing approximations for eigenvalues of a matrix. Test the code on a Hilbert matrix and document the code in a software entry manual.
  7. Task: Use the Rayleigh Quotient algorthm to compute an approximation for the condition of the matrix. Test your code on a Hilbert matrix of reasonable size. Discuss the results from the work.
  8. Task: Look for internet sites that estimate the condition number of a matrix. Document the sites and as usual cite the web pages used in your explanation of the information found.
  9. Task: Compare the Inverse Iteration Algorithm and the Rayleigh Quotient Algorithm in terms of the amount of time needed to compute an eigenvalue. Tabulate your results for the two methods.
  10. Task: Implement a version of the Inverse Iteration Algorithm that uses Jacobi Iteration to compute solutions of the linear systems of equations. Test this algorithm using a random diagonally dominant matrix. Document the algorithm in your software manual.